Phase Stability, Phase Transformations, and Elastic Properties of Cu6 Sn5 : Ab Initio Calculations and Experimental Results G. Ghosh∗ and M. Asta Department of Materials Science and Engineering, Robert R. McCormick School of Engineering and Applied Science, Northwestern University, 2220 Campus Drive, Evanston, IL 60208-3108 Among many Sn-base intermetallics, Cu6 Sn5 (η and η 0 ) is ubiquitous in modern solder interconnects. Using the published structural models of η and η 0 , and also related structures, the total energies and equilibrium cohesive properties are calculated from first-principles employing electronic density-functional theory, ultrasoft pseudopotentials, and both the local density approximation (LDA) and the generalized gradient approximation (GGA) for the exchange-correlation energy. The accuracy of our calculations is assessed through comparisons between theoretical results and experimental measurements for lattice parameters, elastic properties and formation and transformation energies. The ambient-temperature experimental lattice constants of η and η 0 are found to lie between the LDA- and GGA-level calculated zero-temperature lattice constants. The Wyckoff positions in the structural models of η and η 0 agree very well with the ab initio results. The calculated formation energy of η 0 lies between -3.2 to -4.0 kJ/mol which is more positive by about 3 to 4 kJ/mol compared to reported experimental data obtained by solution calorimetry. Our systematic differential scanning calorimetry (DSC) experiments show that the η 0 →η transformation enthalpy is 438±18 J/mol, which is about 66% higher compared to the literature value. In view of our DSC results on heating and cooling, the nature of η 0 →η and η→η 0 is discussed. Our experimental bulk modulus of η and η 0 , and the heat of η 0 →η transformation agree very well with the ab initio total energy calculations at the GGA level. Based on these results, we conclude that other isotropic elastic moduli (Young’s, shear and Poissons ratio) of η and η 0 phases measured by pulse-echo technique are representative of their actual properties. The scatter in experimental elastic constants in the literature may be attributed to various factors, such as the measurement technique (pulse-echo versus nanoindentation), type of specimen (bulk, Cu6 Sn5 -layer in diffusion couple, thin-film), and anisotropy effects (particularly in Cu6 Sn5 -layer in diffusion couples).
I.
INTRODUCTION
Due to the increasingly dense arrays of interconnects, the miniaturization of electronic packages and the use of leadfree solders, there has been a renewed interest in soldering technology. The critical issues of a soldering technology are the wetting behavior and the dynamics of interfacial microstructure evolution during processing and in service.1 Therefore, it is of fundamental and practical interest to quantify the thermodynamic driving forces that govern these phenomena. In addition, the phases that form in solder interconnects during fabrication and under normal service condition
1
Address all correspondence to this author. email:
[email protected]
2 may affect the reliability of the devices through the following intrinsic properties: coefficient of thermal expansion (CTE), ductile/brittle transition, elastic and plastic properties, heat capacity and thermal conductivity. For example, combined CTE, elastic and plastic properties determine the reliability of electronic packages. Both heat capacity and thermal conductivity play important roles in the heat dissipation and extraction processes. Molten solders are known to react with Cu lines very rapidly forming Cu6 Sn5 at the solder/Cu interface.1 In the last fifteen years, numerous publications have appeared dealing with the growth kinetics and morphology of the Cu6 Sn5 intermetallic in solder joints, somewhat surprisingly there has been no research on the phase stability aspects since 1973. There is only one set of calorimetric data for the heat of formation of Cu6 Sn5 ,2 which has been used in the thermodynamic modelling of Cu-Sn phase diagram.3–5 Besides its importance in soldering technology for electronic packaging, Cu6 Sn5 is an attractive alternative to graphite as an anode material in rechargeable lithium-ion batteries.6–8 Lithiation of Cu6 Sn5 leads to the formation of zinc-blend-type phase with a range of stoichiometry, such as Li13 Cu6 Sn5 , Li2 CuSn and Lix Sn.6–8 The advantages of lithiated Cu6 Sn5 are high volumetric capacity compared to lithiated graphite, LiC6 , and lithium does not alloy or react with Cu.6 In electronic packaging, while the formation of a thin intermetallic layer due to the reaction between solder and Cu is desirable to achieve a good metallurgical bond, excessive intermetallic growth may also have a deleterious effect on the mechanical reliability of the solder joints.9–12 Furthermore, due to miniaturization of modern solder joints, the fraction of intermetallic layer compared to the total thickness of solder joint is increasing. Due to these reasons, several attempts have been made to measure the elastic and plastic properties of Cu6 Sn5 using bulk samples,13–16 the interfacial layer in diffusion couples,17–19 and thin films.20 However, a compilation of these data shows relatively large scatter. The structure of Cu6 Sn5 has been repeatedly investigated by X-ray diffraction since 1928, and a brief summary of all results up to 1973 may be found elsewhere.2 There are two forms of Cu6 Sn5 : the high-temperature form (η)2,21–23 is stable above 187 ◦ C and the low-temperature form (η 0 ).2,24–26 In the earlier studies,21,24,25 η was reported to be isotypic with the B81 -NiAs phase. In an ideal NiAs structure the atomic positions are Ni at 2a (0,0,0; 0,0, 21 ) and As at 2c ( 31 , 23 , 14 ;
2 1 3 3 , 3 , 4 ).
However, there may be small displacements of both Ni and As from their ideal positions.
Therefore, a better description of the atomic positions are27 Ni at 2a (0,0,z ; 0,0, 12 +z ) and As at 2b ( 13 , 32 ,z ; 23 , 13 , 12 +z ). Small (and perhaps correlated) displacements of the atoms give rise to many modulated structures, the unit cell of which may have different crystal symmetry, even though it may be derived from B81 -type unit cells. Indeed, Lidin and co-workers,23,26,28 using X-ray and electron diffraction techniques, showed that both η and η 0 have monoclinic symmetry, and they concluded that these are new superstructures belonging to the NiAs-Ni2 In structure group. Based on the X-ray and electron diffraction results, Larsson et al. proposed two structural models for the η phase,23 referred to as η1 and η2 , that correspond to the formula Cu5 Sn4 , and a model for the η 0 phase26 corresponding to the formula Cu6 Sn5 . The prevalence of these structures in many other binary systems was reviewed by Lidin.28,29 A three dimensional perspective of the unit cells of η1 , η2 and η 0 are shown in Fig.1. Here, we present the results of a comprehensive study of zero-temperature energetics, and the equilibrium cohesive properties of η and η 0 phases, and also related structures, using ab initio computational techniques. The crystallographic data of η and η 0 , and other structurally related phases are summarzied in Table I. The goals of this work are: (i) to calculate the structural energetics of η and η 0 phases, in order to augment the calorimetric data for enthalpies of formation and transformation (η 0 ↔η) in support of the development of accurate thermodynamic databases for the
3
C C C
Cu B B
Sn
Sn
O
Cu
Sn Cu B
O
A
A
A
O
(b)
(a)
(c)
Figure 1: Ball-and-stick model of (a) η1 , (b) η2 , and (c) η 0 showing a large number Cu-Sn bonds.
Table I: Crystallographic data of Cu-Sn intermetallics considered in this study. Phase a
Cu2 Sn
Reference
P63 /mmc (194)
hP6 (B82 )
Ni2 In
P63 /mmc (194)
hP4 (B81 )
NiAs
P63 /mmc (194)
hP6 (B82 )
Ni2 In
Cu5 Sn4 (η 1 )
P21 /c (14)
mP36 (...)
Cu5 Sn4
Ref. 23,28
Cu5 Sn4 (η 2 )
C2 (5)
mC54 (...)
Cu5 Sn4
Ref. 23,28
Cu6 Sn5 (η’)
C2/c (15)
mC44 (...)
Cu6 Sn5
Ref. 26,28
CuSn CuSn2
a Virtual
Space Group (#) Pearson Symbol (SS) Prototype
a
Ref. 2,21,24,25
compound
Cu-Sn system, and their elastic properties by ab initio techniques, (ii) to experimentally verify the ab initio predictions of the heat of η 0 ↔η transformation and the bulk moduli of η and η 0 phases, and (iii) to rationalize the scatter in experimental elastic property data reported in literature.
II. A.
COMPUTATIONAL METHODOLOGY Ab Initio Total Energy Calculations
The ab initio calculations presented here are based on electronic density-functional theory (DFT), and have been carried out using the ab initio total-energy and molecular-dynamics program VASP (Vienna ab initio simulation package) developed at the Institut f¨ ur Materialphysik of the Universit¨at Wien.30,31 The current calculations make use of the VASP implementation of Vanderbilt-type ultrasoft pseudopotentials (US-PP),32 and an expansion of the electronic wavefunctions in plane waves with a kinetic-energy cutoff of 314 eV. The US-PPs employed in this work explicitly treat eleven valence electrons for Cu (3d10 4s1 ) and fourteen for Sn (4d10 5s2 p2 ). All calculated results were derived employing both the local-density approximation (LDA), with Ceperly-Alder33 form of exchange-correlation (xc) energy as param-
4 eterized by Perdew and Zunger,34 and the generalized gradient approximation (GGA) for exchange-correlation energy due to Perdew and Wang.35 Brillouin-Zone integrations were performed using Monkhorst-Pack36 k-point meshes, and the Methfessel-Paxton37 technique with a 0.1 eV smearing of the electron levels. For each structure, tests were carried out using different k-point meshes to ensure absolute convergence of the total energy to within a precision of better than 1.0 meV/atom (0.1 kJ/mole). As an example, the k-meshes for η and η 0 structures were 5 × 7 × 5 and 5 × 8 × 6, respectively. Depending on the structure, up to 264 k-points were used in the irreducible Brillouin zone. We have used at least 0.001 meV/atom as a criterion for the self-consistent convergence of total energy. In the case of η 1 , η 2 and η 0 phases, we have used the experimetally determined structural parameters as input to ab initio calculations. For all structures considered, we have calculated the total energies as a function of volume while optimizing unit cell-external degree(s) of freedom (i.e., the unit-cell shape) and unit cell-internal degree(s) of freedom (i.e., Wyckoff positions) as permitted by the space-group symmetry of the crystal structure. Such structural optimizations were iterated until the Hellman-Feynman forces were less than 4 meV/˚ A in magnitude, ensuring a convergence of the energy with respect to the structural degrees of freedom to better than 1 meV/atom (0.1 kJ/mole). With the chosen plane-wave cutoff and k-point sampling the reported formation energies are estimated to be converged to a precision of better than 2 meV/atom (0.2 kJ/mole). All results presented below were obtained employing the computational settings described in the previous paragraph. In addition, several calculations at LDA and GGA levels for pure Sn, and at GGA level for Cu2 Sn, CuSn and CuSn2 were also performed with alternative settings to gauge the overall accuracy of the reported results. Specifically, additional calculations were performed employing an alternative US-PP for Sn that did not include semi-core 4d states as valence i.e., only four electrons (5s2 p2 ) were explicitly treated as valence. As will be shown below, this led to decreases in the calculated formation energies of intermetallics (i.e., more negative values) by only about 0.6 kJ/mole, and changes the computed lattice constants by about 0.2 percent.
B.
Equation of State and Formation Energy
We take the zero-temperature formation energy (∆E φ ) of an intermetallic Cum Snn , where m and n are integers, as a key measure of the relative stability of competing structures (φ1 , φ2 , φ3 ...). The formation energy per atom is evaluated relative to the composition-averaged energies of the pure elements in their equilibrium crystal structures: ∆E φ (Cum Snn ) =
1 m n φ θ ECu ECu + Eψ ] −[ m Snn m+n m+n m + n Sn
(1)
φ θ where ECu is the total energy of Cum Snn with structure φ, ECu is the total energy per atom of Cu with the fcc m Snn ψ (θ) structure and ESn is the total energy per atom of Sn with the A5-tetragonal (ψ) structure.
The equation of state (EOS) generally defines the relationship between pressure (P ), volume (V ) and temperature (T ). Here we consider only zero-temperature equations of state, defining pressure-volume relationships. Numerous forms for such EOS can be found in the literature, and, in fact, the search for a universal form of the EOS of solids is still an important problem in high-pressure physics and geophysics. We have used the EOS due to Vinet et al.38 who assumed the interatomic interaction-versus-distance relation in solids can be expressed in terms of a relatively few material constants. The most commonly used EOSs given by Murnaghan39 and Birch40 work as well as that by Vinet et al.38 at low pressures, but at ultra-high pressure Birch-Murnaghan EOSs, which are based on lower-order
5 Taylor-series expansions, are known to be less accurate. In the EOS of Vinet et al.38 the pressure P is expressed in terms of isothermal bulk modulus (Bo ), its pressure derivative (Bo0 ) and a scaled quantity (x ): P = 3Bo x−2 (1 − x) exp[χ(1 − x)]
(2)
with x = (V /Vo )1/3 and χ = 3/2(Bo0 − 1), where Vo is the equilibrium volume. Based on Eq. (2) and the relations between pressure and energy, the total energy (E ) and volume-dependence of the bulk modulus can be expressed as E(V ) − E(Vo ) =
9Bo Vo {1 − [χ(1 − x)] exp[χ(1 − x)]} χ2
B(V ) = x−2 [1 + (χx + 1)(1 − x)] exp[χ(1 − x)] Bo
(3)
(4)
Vinet et al.38 have shown that the second-order pressure derivative of the bulk modulus , which is a more severe test of the accuracy of EOS, can be expressed as: Bo Bo00 =
19 1 0 1 − Bo − (Bo00 )2 36 2 4
(5)
Equations (2 to 5) are found to work well for metallic, covalent, ionic and van der Waals bonded solids.38
III.
EXPERIMENTAL PROCEDURE
High purity (at least 99.99%) elements were used to prepare the alloy. Since Cu6 Sn5 forms by a peritectic reaction, single phase bulk specimens were prepared by conventional casting followed by annealing for long times. Specifically, to obtain η phase solid state annealing was performed at 222 ◦ C for 72 days after casting. To obtain η 0 phase, a part of the ingot that was annealed at 222 ◦ C was further annealed at 150 ◦ C for 80 days, two parts of which were further annealed at 100 and 125 ◦ C for 60 days. Structural and microstructural characterization of the annealed specimens were carried out by X-ray diffraction and scanning electron microscopy (SEM). X-ray diffraction, using powder specimens, was performed using a Scintag machine (Scintag, Inc., CA, USA) equipped with a copper target, excited to 40kV and 20mA, and a solid state detector. A Hitachi H-3500 SEM (Hitachi, Ltd., Japan) operated at 20 kV was used to determine the microstructure. The temperature of the η↔η 0 transformation of Cu6 Sn5 and the associated enthalpy of transformation were determined by differential scanning calorimetry (DSC-TSO801R0, Mettler-Toledo, Schwerzenbach, Switzerland). Both temperature and heat flow were calibrated using pure (99.99%) indium. Constant heating/cooling rates of 1, 2, 5 and 10 ◦ C per minute were used. The isotropic elastic moduli were measured by an ultrasonic pulse-echo technique. Unlike plastic deformation experiments, such as compression, tensile and shear testing, to determine the elastic moduli, the measurement of sound velocities in solids yields accurate values of elastic constants without inducing any plastic deformation. This method utilizes wave mechanics theory that relates the sound wave velocities and elastic moduli.41–43 We have used an SR 9000 Ultrasonic Modulus Tester (Matec Instruments, Hopkinton, MA, USA) to measure the longitudinal and shear wave velocities. Cylindrical specimens of about 3 mm thick were cut from the annealed ingots, and both flat surfaces were subjected to a 1 µm polish. Axially-polarized and cylindrically-polarized transducers generated 50 MHz
6 longitudinal and 20 MHz shear stress pulses, respectively. The reflections of the stress pulse from the front and back surfaces of the specimens were continuously monitored by a computer. The software system allowed the reflections to be frozen in the computer at any time from which the transit time (round trip) for reflections was measured and the velocity of longitudinal (ϑl ) and shear waves (ϑs ) were determined. Then, the isotropic shear modulus (µ), the Young’s modulus (E), the bulk modulus (B), and the Poissons ratio (ν) are given by µ = ρϑ2s
(6)
E = 2(1 + ν)µ
(7)
Eµ 3(3µ − E)
(8)
B=
2−R withR = ν= 2(1 − R)
ϑl ϑs
2 (9)
For each specimen, an average of four measurements of ϑl and ϑs was taken to calculate the elastic constants. Ambient temperature theoretical densities (ρ) of η and η 0 , based on their lattice parameters,28 are evaluated as 8448 kg/m3 and 8270 kg/m3 , respectively.
IV. A.
RESULTS
Ab initio results: Lattice stability and cohesive properties of Cu and Sn
Lattice stability of pure elements is very important in CALPHAD44 (CALculation of PHAse Diagrams) modelling of phase diagrams that begins with the Gibbs energy description of pure elements with different structures. We have calculated the total energies of Cu and Sn as a function of volume for the following structures: fcc (A1), bcc (A2), hcp (A3), diamond cubic (A4), and tetragonal (A5 and A6). The resulting zero-temperature cohesive properties are compared with available experimental data in Table II. The lattice stabilities at 0 K, defined by relative structural energies, are listed in Table III.
Table II: Calculated structural and elastic properties of Cu and Sn at 0 K. The units of lattice parameter, V ◦ , B ◦ are nm, nm3 /atom and 1010 N/m2 , respectively.
EOS parameters xc LDA
Element Space group (#) Pearson symbol (SS) Cu
GGA LDA
Sn
Im3m (229)
cI2 (A2)
Prototype
Lattice parameters
V◦
B◦
B 0◦
W
a=0.28061
0.011048
18.37
5.47
a=0.28958
0.012142
14.33
6.37
a=0.37082
0.025496
6.05
3.35
(a=0.37008
0.025342
6.02
4.74)a
Continued in next page. . . . . . . . . . . . . . . . . .
7 Table II: Table II (continued from previous page) EOS parameters xc
Element Space group (#) Pearson symbol (SS)
Prototype
GGA
LDA
Cu
F m3m (225)
cF 4 (A1)
Cu
GGA
V◦
B◦
(a=0.37529
0.026430
4.97)b
(a=0.37330
0.026010
5.28)c
a=0.38124
0.027706
4.84
4.72
(a=0.38055
0.027555
4.80
4.72)a
a=0.35301
0.010998
19.02
4.90
0.012093
13.91
2.42
a=0.36436 (a=0.35956)
LDA
Sn
GGA
LDA
Cu
P 63 /mmc (194)
hP 2 (A3)
Mg
GGA LDA
Sn
GGA
LDA
Cu
F d3m (227)
cF 8 (A4)
C (diamond)
GGA LDA
Sn
GGA
d
e
Cu
GGA LDA
Sn
I41 /amd (141)
tI4 (A5)
β-Sn
(14.2)
(5.48,5.04)f
a=0.46939
0.025855
5.64
5.28
(a=0.46855
0.025716
5.83
4.57)a
(a=0.47587
0.026940
5.08)b
(a=0.47350
0.026540
5.14)c
a=0.48255
0.028096
4.67
5.29
(a=0.48171
0.027945
4.79
3.98)a
a=0.24961, c=0.40837 0.011010
18.63
4.72
a=0.25761, c=0.42117 0.012118
13.62
5.01
a=0.32660, c=0.55528 0.025636
5.62
4.13
(a=0.32847, c=0.54422 0.025440
5.91
4.69)a
(a=0.33570, c=0.54652 0.026670
6.16)b
(a=0.33220, c=0.54910 0.026240
5.13)c
a=0.33629, c=0.56986 0.027893
4.72
4.59
(a=0.33626, c=0.56503 0.027665
4.74
4.65)a
a=0.52016
0.017216
7.50
4.90
a=0.53634
0.019286
5.59
5.27
a=0.64748
0.033931
4.67
4.14
(a=0.64549
0.033619
4.47
4.79)a
(a=0.65472
0.035080
4.47)b
(a=0.64620
0.033720
4.32)c
a=0.66487
0.036738
3.66
4.66
(a=0.66317
0.036458
3.59
4.92)a
(a=0.67380
0.038240
3.08)b
(a=0.64804)g
LDA
B 0◦
Lattice parameters
5.3,h 5.4i
a=0.45097, c=0.23778 0.012082
15.18
4.74
a=0.46619, c=0.24541 0.013329
11.27
3.71
a=0.57798, c=0.31338 0.026173
5.98
5.03
Continued in next page. . . . . . . . . . . . . . . . . .
8 Table II: Table II (continued from previous page) EOS parameters xc
Element Space group (#) Pearson symbol (SS)
Prototype
GGA
B◦
B 0◦
(a=0.57749, c=0.31229 0.026043
6.01
4.72)a
(a=0.58647, c=0.31728 0.027280
5.44)b
(a=0.58000, c=0.31410 0.026420
5.53)c
a=0.59469, c=0.32177 0.028443
4.60
4.33
(a=0.59340, c=0.32138 0.028295
4.72
4.33)a
(a=0.60250, c=0.32560 0.029550
4.14)c
Lattice parameters
V◦
(a=0.58015, c=0.31474)j
LDA
Cu
GGA LDA
GGA
a b c d e f g h i j k
Sn
I4/mmm (139)
tI2 (A6)
In
(5.79)k (6.01,4.96)f
a=0.28656, c=0.26916 0.011050
18.59
4.69
a=0.29436, c=0.28040 0.012151
13.56
3.91
a=0.39049, c=0.33563 0.025605
5.89
6.28
(a=0.39025, c=0.33437 0.025474
5.86
4.44)a
(a=0.38905, c=0.35014 0.026490
5.20)b
(a=0.39820, c=0.34170 0.027090
5.48)c
a=0.40333, c=0.34226 0.027861
4.52
4.62
(a=0.40263, c=0.34177 0.027701
4.64
4.97)a
This study using alternate US-PP description with four valence electrons (5s2 p2 ) in Sn FP-LMTO, LDA45 NC-PP, LDA46 Based on experimental data47,48 extrapolated to 0 K. Experimental data at 0 K.49,50 Experimental data at 298 K.51 Based on experimental data52 extrapolated to 0 K. Low-temperature data53 Low-temperature data54 Based on experimental data55,56 extrapolated to 0 K. Experimental data at 0 K.50,57
The quoted experimental lattice parameters of fcc-Cu,47,48 A4-Sn,52 (also called α-Sn or grey tin) and A5-Sn55,56 (also called β-Sn or white tin) at 0 K are obtained by extrapolation of corresponding measurements at higher temperatures to 0 K. The experimental bulk moduli of A1-Cu49,50 and A5-Sn50,57 in Table II are also based on extrapolation to 0 K. The experimental B 0◦ values of A1-Cu and A5-Sn are at 298 K.51 It has been pointed out51 that depending on the measurement technique, ultrasonic resonance, versus the initial slope of the locus of Hugoniot states in shock-velocity particle-velocity coordinates, the value of B 0◦ may differ even though ideally they should be the same. In the case of A1-Cu, the experimental lattice parameter and bulk modulus agree within 1.3% and 1%, respectively, when compared with the corresponding values obtained by GGA. As expected, the US-PP-LDA values manifest the effect of overbinding causing smaller lattice parameter and higher bulk modulus. These results are consistent with the previously reported values59 based on FLAPW-LDA (Full-Potential Linearized Augmented Plane Wave). Wang and
9 ˇ 60 noted that depending on the treament of exchange-correlation energy, the lattice parameter of A1-Cu varies Sob from 0.352 to 0.361 nm and its bulk modulus varies from 15.3 to 19×1010 N/m2 , while the corresponding values for A2-Cu varies from 0.28 to 0.287 nm, and 16 to 18.8×1010 N/m2 . The calculated lattice parameters of A6-Cu based on US-PP-LDA and US-PP-GGA are within 2% of the previously reported values59 based on FLAPW-LDA and FLAPW-GGA, respectively. Previous ab initio studies of phase stability of Sn have been carried out by the full-potential linear muffin tin orbital (FP-LMTO) method45 and using norm-conserving pseudopotentials (NC-PP).46,58,61 All of these previous calculations made use of LDA. Besides our results, there is only one instance where calculations were performed at both LDA and GGA levels.46 NC-PP-based calculations by Aguado46 found that A4-Sn is the ground state for both LDA and GGA calculations. On the other hand, as seen in Table III, VASP-GGA (with and without semicore treatment) correctly predicts the ground state to be A4, while VASP-LDA (with and without semicore treatment) predicts it to be A5. In the case of both A4- and A5-Sn, the experimental lattice parameter and bulk modulus agree within 2.4% and 25 to 40%, respectively, when compared with the corresponding values obtained by our GGA calculations as well as those based on NC-PP-GGA.46 On the other, the experimental lattice parameters, B o and B 0o values show a very good agreement with the LDA values, despite the fact that VASP-LDA (with and without semicore correction) fails to predict the correct ground state of Sn. Like our results, previously reported LDA-based results45,46,58,61 also show good agreement with the experimental lattice parameter and bulk modulus data for both A4- and A5-Sn. The importance of treatment of the semicore 4d-states as valence on the phase stability and cohesive properties of Sn has been discussed in detail by Christensen and Methfessel.45 Explicit modeling of the 4d-states of Sn as semicore electrons, accounting in an approximate way for self-interaction correction, was reported necessary to obtain the correct transformation sequence A4→A5→A6→A2, as has been observed experimentally, as a function of pressure. On the other hand, Aguado46 generated NC-PP of Sn with four electrons as valence, and obtained the same sequence of phase transitions and at pressures that agree very well with the experimental values. Furthermore, Aguado46 emphasized that LDA provides good agreement with the structural and elastic properties and phase transition sequence, while GGA accurately reproduces only the experimental binding energies and not the phase transition sequence. In the present study, the effect of pressure on the phase transition of Sn was not investigated; however, comparing the cohesive properties of Sn based on the GGA, listed in Table II, we find that the US-PPs with and without semi-core treatment give very similar results (usually within 1%) for the lattice parameters and bulk modulus and its pressure derivative. The calculated lattice stabilities of Cu and Sn at 0 K are compared with those from the SGTE (Scientific Group Thermo-Data Europe) database,62 as provided in Thermo-Calc software package,63 in Table III. Our calculated lattice stabilities of Cu and Sn involving bcc, fcc and hcp structures based on US-PP-GGA, listed in Table III, are in very good agreement (within 0.5 kJ/mol) with recently reported64 VASP-GGA calculations based on the ProjectorAugmented-Wave (PAW) method.65 The first-principles structural energy differences can be sensitive to both the A1→A2 DFT method and the treatment exchange-correlation energy. For example, compilation of ∆E Cu first-principles
data60 show a scatter of 655-4720 J/mol. For simple structures, such as bcc, fcc and hcp, the quantitative differences between our calculated values and SGTE database are less than 1 kJ/mol, thus the agreement should be considered as good. However, in the case of Sn, we note that the SGTE database predicts that A5 structure is more stable compared to A4 at very low temperatures; this unexpected lattice stability for Sn in the SGTE database could be an
10 Table III: A comparison of lattice stabilities (in J/mol) of Cu and Sn. For direct comparison with the calculated results, lattice-stability values from the SGTE database62 are all reported at 0 K. Ab initio (This study)
Ab initio (Previous studies)
Property
xc-LDA
xc-GGA
∆E A1→A2 Cu
3863.7
3343.4
xc-LDA
xc-GGA
SGTE Database62
2861.0a
4017
∆E A1→A3 Cu
1098.4
982.8
∆E A1→A4 Cu
119332.8
97162.1
b
∆E A1→A5 Cu
36238.3
30312.6
b
∆E A1→A6 Cu
3863.7
3430.2
b
∆E A5→A1 Sn
4721.3
3131.5 d
∆E A5→A2 Sn
∆E A5→A6 Sn
5963.6c d
7226.5
4673.1
4066.1
6399.4c d
4400
4150
e
(4673.1)
(3998.6)
6744.7
3883.0
2823.1
5367.6c d
600
e
(3092.9)
d
∆E A5→A4 Sn
520.3
(4759.8)
d
∆E A5→A3 Sn
a
3900
e
(3921.6)
(2948.4)
7033.7
2139.0
-4384.0
-943.4c
(1724.7)d
(-4557.5)d
-3372.3e
3805.9
3015.0
5687.9c
(3825.2)d
(2967.7)d
3372.3e
f
-2601.5e b
a PAW64 b Not
provided
c FP-LMTO45 d Based
on alternate US-PP description for Sn with four valence electrons (5s2 p2 )
e NC-PP46 fA
large and positive number
artifact of their functional representation, since the correct lattice stability of Sn is predicted at room temperature. Neither Dinsdale62 (also SGTE database) nor Wang et al.64 reported the lattice stabilities of Cu and Sn in A4, A5, and A6 structures. Their lattice stabilities are important in thermodynamic modelling of Sn-based binary and ternary phase diagrams. The calculated lattice stabilities of these structures based on VASP-LDA and VASP-GGA are compared with the previously reported values based on FP-LMTO-LDA45 and NC-PP-LDA.46 In the case of Sn, we note that depending on the method, the lattice stability values differ by up to about 3.3 kJ/mol. We also find that for both LDA and GGA level calculations, the lattice stabilities of Sn change by less than 0.5 kJ/mol when 4d-states of Sn are not considered as valence.
B.
Ab initio results: Phase stability and cohesive properties of Cu6 Sn5
To obtain the equilibrium geometry of the model intermetallic structures for Cu6 Sn5 ,23,26,28 and their equilibrium cohesive properties, the following procedure was adopted. First, taking the experimental unit cell data28 of η1 , η2 and η 0 as input, the cell-internal and cell-external parameters were globally optimized by minimizing the stresses and
11 interatomic forces. Then, the total energy was calculated as a function of volume while relaxing the cell-internal and cell-external parameters at fixed volume. The energy-volume data was then fit with the EOS in Eq. (3). Figure 2 shows the E-V plots defining zero-temperature EOS parameters for η and η 0 phases based on VASP-LDA and VASPGGA calculations. All results of ab initio calculations for Cu-Sn intermetallics are summarized in Tables IV, V, and VI.
(a)
(b)
Figure 2: Calculated zero-temperature total energy as a function of volume, E(V ), for η1 , η2 , and η 0 phases: (a) US-PP-LDA and (b) US-PP-GGA. The symbols represent calculated values, and the line is a fit to EOS in Eq. (3) .
Comparing the Vo and Bo values in Table IV, and the lattice parameters in Table V we find that the differences between LDA and GGA results are consistent with the expected trend (namely LDA gives rise to smaller values of lattice parameters and higher elastic moduli relative to GGA). In the earlier structural studies of the high-temperature form of Cu6 Sn5 2,21,25 , the X-ray diffraction peaks were indexed as B81 -CuSn. In Table V, it is seen that the reported lattice parameters of B81 -CuSn are in very good agreement with the GGA (with and without semicore treatment of Sn) rather than the LDA values. However, this may be just a coincidence as the actual compositions are different, 50 at.% Sn in the calculation versus 45 at.% Sn in the experimental alloy; for example in going from B81 -CuSn to B82 -Cu2 Sn, the calculated values of a and c parameters increase (see Table V). On the other hand, for the η1 , η2 and η 0 phases, where calculations are carried out at the same composition as the actual phase composition, the experimental lattice parameters lie between the LDA and GGA calculated values. As seen in Table V, the phases with B82 structures, Cu2 Sn and CuSn2 , are not thermodynamically stable as their formation energies (∆E f ) are very positive. The ∆E f of the thermodynamically stable intermetallics, B81 -CuSn, η1 , η2 and η 0 , calculated by LDA and GGA agree within 1 kJ/mol (see Table V). However, our calculated ∆E f of η 0 is about 3 to 4 kJ/mol more positive compared to a previously reported value obtained by solution calorimetry.2 Since all CALPHAD assessments3–5 are based on this experimental value, they all agree very well with each other. Even though there is no measured ∆E f for the η phase, in CALPHAD assessments it is based on the previously reported enthalpy of η 0 →η transformation.2
12 Table IV: Equilibrium cohesive properties at 0 K, as defined by the equation of state, of Cu-Sn intermetallics. The units of V ◦ and B ◦ are nm3 /atom and 1010 N/m2 , respectively. EOS parameters xc-LDA
a
xc-GGAa
Phase
Space group (#)
V◦
B◦
B 0◦
V◦
B◦
B 0◦
Cu2 Sn
P63 /mmc (194)
0.014912
11.86
5.14
0.016424
8.82
4.38
(0.016352
8.80
4.88)b
0.019459
7.65
5.10
(0.019350
7.65
5.14)b
0.023530
4.94
3.97
(0.023406
4.94
4.92)b
CuSn
P63 /mmc (194)
CuSn2
P63 /mmc (194)
0.017764
0.021530
10.18
6.19
5.18
3.01
Cu5 Sn4 (η 1 )
P21 /c (14)
0.016684
10.76
5.04
0.018282
8.17
4.98
Cu5 Sn4 (η 2 )
C2 (5)
0.016716
10.62
5.13
0.018323
8.15
4.98
c
8.46 Cu6 Sn5 (η’)
C2/c (15)
0.016887
10.59
5.10
0.018512
7.96
5.02
c
8.44 a This
study on alternate US-PP description for Sn with four valence electrons (5s2 p2 ) c Experimental data at 298 K
b Based
C.
Experimental results: Phase transformations and elastic properties of Cu6 Sn5
η↔η 0 Phase transformations: Figures 3(a) and 3(b) show the X-ray powder diffraction patterns of η (annealed at 220 ◦
C) and η 0 (annealed at 150 ◦ C), respectively. Since our objectives were to determine the heat of η↔η 0 transformation
and to measure the effect of the structural transformation on the elastic properties, if any, it was necessary to confirm that we indeed obtained the η 0 phase after annealing η at 100, 125 and 150 ◦ C. In Fig. 3(b), it is interesting to note the presence of several relatively weak peaks, in the 2θ range of 25 to 50◦ , in addition to those for the η phase shown in Fig. 2(a). For both η and η 0 , the diffraction peaks are indexed according to monoclinic symmetry models proposed by Lidin and Larsson.28 Figure 4 shows the DSC heat flow curves of Cu6 Sn5 in as-cast specimen and after annealing treatments. On heating, while the as-cast specimen shows one asymmetric and rather weak peak (see Fig. 4(a)), the annealed specimens show one symmetric and stronger peak (see Figs. 4(b) to 4(d)). However, on cooling the annealed specimens yielded an exothermic peak that is asymmetric, suggestive of overlapping peaks due to two phase transformations. In DSC thermograms, the onset temperature of a phase transformation is determined by the intersection of the baseline and the tangent of steepest descent. Irrespective of low temperature annealing treatment and heating rate (but within 1 to 10 ◦ C/min), our DSC results give the η 0 →η transformation temperature to be 187 ± 1 ◦ C which is also the value reported by Gangulee et al.2 However, we find that on cooling the η→η 0 onset temperature decreases with increasing cooling rate, from 186 ◦ C at 1 ◦ C/min to 180 ◦ C at 10 ◦ C/min. Gangulee et al.2 did not report any such result. The area under the DSC peaks on heating defines the latent heat associated with η 0 →η transformation, and that of η→η 0 transformation on cooling. These values are determined using five specimens of each annealing treatment
13
(a)
(b)
Figure 3: Indexed X-ray powder diffraction pattern of (a) η-Cu6 Sn5 and (b) η 0 -Cu6 Sn5 .
that were heated/cooled at 1 or 5 ◦ C/min in DSC. These results are summarized in Table VII, and are compared with our ab initio predictions. In as-cast Cu6 Sn5 the heat of η 0 →η transformation is only 13.8 J/mol. On the other hand, the endothermic peaks of the annealed specimens are much stronger with the heat of transformation in excess of 500 J/mol. These results obviously imply incompleteness of η→η 0 transformation due to fast cooling of the as-cast specimen. Furthermore, our experimental data in Table VII demonstrate that (i) the endothermic heat associated with η 0 →η transformation decreases with increasing annealing temperature, particularly at 150 ◦ C or above, and (ii) the absolute value of heat of η 0 ↔η phase transformation is less on cooling than on heating. Based on the results obtained using 100 ◦ C and 125 ◦ C annealed specimens, we take the transformation enthalpy of η 0 →η as 438±18 J/mol. This is about 66% more than the value, 263.7±33 J/mol, determined by solution calorimetry technique.2 Once again, all CALPHAD assessments3–5 are based on the heat of transformation reported by Gangulee et al.2 , hence, they all agree very well with each other.
14 Table V: A comparison of unit cell-external parameters (in nm) of Cu-Sn intermetallics obtained by ab initio calculations (at 0 K) and diffraction experiments (at ambient temperature). Also compared is the heat of formation (∆E f , in J/mol) of intermetallics obtained by various methods: ab inito calculations (at 0 K), experiment, and CALPHAD (at 298.15 K) using the reference states of fcc-Cu (A1) and bct-Sn (A5). Unit cell-external parameter a
Phase
Space group (#)
xc-LDA
Cu2 Sn
P63 /mmc (194)
a=0.43874
a
xc-GGA
∆E f
Experimental xc-LDA
a=0.45329
c=0.53668
a
16752.5
c=0.55434) P63 /mmc (194)
a=0.40759
P63 /mmc (194)
Experimental CALPHAD
14472.2 (13922.9)b
b
a=0.41981
c=0.49418
CuSn2
xc-GGA
c=0.55419 (a=0.45230
CuSn
a
c=0.51003
(a=0.41900 c=0.50860)
(a=0.41899
(a=0.41920
c=0.50917)b
c=0.50370)2
a=0.43646
c=0.44899
c=0.78350
c=0.80823
-5458.4
-4485.2
21
(-5125.9)b
19803.7
15454.9 (15069.6)b
(a=0.44787 c=0.80888)b Cu5 Sn4 (η 1 )
Cu5 Sn4 (η 2 )
Cu6 Sn5 (η 0 )
a This
P21 /c (14)
C2 (5)
C2/c (15)
a=0.97524
a=1.00573
a=0.98328
b=0.71949
b=0.74157
b=0.727
-3665.7
-2884.1
-3270.7
-2595.1
c=0.97028
c=1.00011
c=0.983
β = 61.89◦
β = 61.89◦
β = 62.5◦
a=1.25149
a=1.29139
a=1.26028
b=0.72052
b=0.74202
b=0.727
-70864
c=1.00137
c=1.03250
c=1.020
-74445
β = 89.59◦
β = 89.56◦
β = 90◦
a=1.08002
a=1.11377
a=1.103628
b=0.71786
b=0.74027
b=0.7288
-73464
c=0.96907
c=0.99948
c=0.9841
-77485
β = 98.58◦
β = 98.65◦
β = 98.81◦
-4019.8
-3205.0
-68703
-7037±592
-71303
study on alternate US-PP description for Sn with four valence electrons (5s2 p2 )
b Based
In addition to single heating/cooling experiments, cyclic experiments were also carried out in DSC using the same specimen. In such experiments, specimens were heated and cooled, four times each, between 100 and 220 ◦ C and at a rate of 1 ◦ C/min. As an example, Fig. 5 shows the DSC thermograms, after cyclic experiment of a specimen that was annealed at 100 ◦ C. The heat of transformation associated with these thermal cycles is listed in Table VII. It is seen that the magnitude of the heat of transformation decreases after each cycle, implying the incompleteness of η→η 0 transformation after each cycle due to sluggish kinetics. However, the transformation temperatures are not affected appreciably due to thermal cycles.
15 Table VI: A comparison of unit cell-internal parameters (Wyckoff positions) of η and η 0 obtained from our ab initio calculations (at 0 K) and diffraction experiments (at ambient temperature). Unit cell-internal parameters Phase
Space group (#)
Cu5 Sn4 (η1 )
P21 /c (14)
Cu5 Sn4 (η2 )
Cu6 Sn5 (η 0 )
C2 (5)
C2/c (15)
Site
(x, y, z ): xc-LDA
(x, y, z ): xc-GGA
(x, y, z ): Experiment
Cu1: 4e 0.05605, 0.77549, 0.05692 0.05595, 0.77598, 0.05695 0.057, 0.779, 0.05623,28 Cu2: 4e 0.18939, 0.25523, 0.18691 0.18954, 0.25562, 0.18691
0.189, 0.253, 0.187
Cu3: 4e 0.19119, 0.08864, 0.93804 0.19124, 0.08868, 0.93830
0.188, 0.090, 0.937
Cu4: 4e 0.31559, 0.72496, 0.31663 0.31588, 0.72416, 0.31705
0.317, 0.725, 0.321
Cu5: 4e 0.42976, 0.24361, 0.43156 0.42967, 0.24260, 0.43147
0.427, 0.258, 0.427
Sn1: 4e 0.07108, 0.59749, 0.28467 0.07119, 0.59795, 0.28461
0.071, 0.597, 0.287
Sn2: 4e 0.18365, 0.05282, 0.43553 0.18332, 0.05260, 0.43539
0.180, 0.057, 0.440
Sn3: 4e 0.32424, 0.58539, 0.57694 0.32456, 0.58491, 0.57739
0.313, 0.590, 0.583
Sn4: 4e 0.45600, 0.09452, 0.66881 0.45611, 0.09517, 0.66888
0.451, 0.087, 0.663
Cu1: 2a 0.00000, 0.67917, 0.00000 0.00000, 0.67899, 0.00000 0.000, 0.691, 0.00023,28 Cu2: 4c 0.00857, 0.01371, 0.37659 0.00839, 0.01309, 0.37677
0.015, 0.030, 0.374
Cu3: 4c 0.00749, 0.99666, 0.12554 0.00755, 0.99615, 0.12559
0.022, 0.007, 0.127
Cu4: 4c 0.17026, 0.85626, 0.25080 0.17041, 0.85624, 0.25091
0.173, 0.853, 0.254
Cu5: 4c 0.32556, 0.01218, 0.37668 0.32553, 0.01276, 0.37684
0.326, 0.000, 0.379
Cu6: 4c 0.33402, 0.01909, 0.12559 0.33379, 0.01933, 0.12565
0.343, 0.017, 0.129
Cu7: 4c 0.66011, 0.04152, 0.12555 0.66031, 0.04199, 0.12547
0.653, 0.034, 0.133
Cu8: 4c 0.66799, 0.03566, 0.37649 0.66809, 0.03573, 0.37669
0.666, 0.036, 0.379
Sn1: 2a 0.00000, 0.32235, 0.00000 0.00000, 0.32198, 0.00000
0.000, 0.333, 0.000
Sn2: 2b 0.00000, 0.34987, 0.50000 0.00000, 0.34969, 0.50000
0.000, 0.360, 0.500
Sn3: 4c 0.16389, 0.21382, 0.24902 0.16336, 0.21398, 0.24907
0.179, 0.210, 0.249
Sn4: 4c 0.32089, 0.36511, 0.01874 0.32089, 0.36534, 0.01873
0.307, 0.349, 0.009
Sn5: 4c 0.33165, 0.35612, 0.48042 0.33152, 0.35647, 0.48046
0.339, 0.352, 0.477
Sn6: 4c 0.49043, 0.17048, 0.26866 0.49062, 0.16971, 0.26856
0.488, 0.190, 0.268
Sn7: 4c 0.84945, 0.17767, 0.24933 0.84971, 0.17788, 0.24944
0.845, 0.164, 0.240
Cu1: 4a 0.00000, 0.00000, 0.00000 0.00000, 0.00000, 0.00000 0.000, 0.000, 0.00026,28 Cu2: 4e 0.00000, 0.16081, 0.25000 0.00000, 0.16101, 0.25000
0.000, 0.160, 0.250
Cu3: 8f 0.10136, 0.47345, 0.20347 0.10165, 0.47283, 0.20396
0.101, 0.473, 0.202
Cu4: 8f 0.30659, 0.50426, 0.60948 0.30668, 0.50479, 0.60941
0.306, 0.504, 0.610
Sn1: 4e 0.00000, 0.80012, 0.25000 0.00000, 0.80004, 0.25000
0.000, 0.799, 0.250
Sn2: 8f 0.28651, 0.65568, 0.35763 0.28645, 0.65539, 0.35741
0.285, 0.655, 0.358
Sn3: 8f 0.39198, 0.16277, 0.52908 0.39184, 0.16307, 0.52858
0.391, 0.162, 0.529
Elastic properties of η and η 0 phases: The measured bulk modulus, shear modulus, Young’s modulus and Poissons ratio for the η phase are 8.46 × 1010 N/m2 , 3.70 × 1010 N/m2 , 9.69 × 1010 N/m2 , and 0.309, respectively. The corresponding values for the η 0 phase are 8.44×1010 N/m2 , 3.59×1010 N/m2 , 9.44×1010 N/m2 , and 0.314, respectively.
16
(a)
(b)
(c)
(d)
Figure 4: DSC thermograms of (a) as-cast Cu6 Sn5 heated at 5 ◦ C/min, (b) η 0 -Cu6 Sn5 , annealed at 150 ◦ C for 80 days followed by 100 ◦ C for 60 days, and then heated/cooled at 1 ◦ C/min, (c) η 0 -Cu6 Sn5 , annealed at 125 ◦ C for 80 days followed by 125 ◦ C for 60 days, and then heated/cooled at 1 ◦ C/min, and (d) η 0 -Cu6 Sn5 , annealed at 150 ◦ C for 80 days, and then heated/cooled at 2 ◦ C/min .
Taking high-purity polycrystalline Fe and single crystals of Cu as standards, we estimate that the uncertainty of our measured values of elastic moduli is ±5% at 95% confidence level. The measured bulk moduli of η and η 0 phases at ambient-temperature may be compared with the corresponding ab initio values at 0 K in Table IV. The important results to be noted are: (i) VASP-GGA values agree within the experimental uncertainty mentioned above, while the VASP-LDA values are considerably higher, (ii) ab initio results show that the predicted bulk modulus of the η phase is virtually the same irrespective of the structural models (i.e.,
17 Table VII: A comparison of heat of transformation (∆E tr ) obtained from our ab initio calculations (at 0 K) and the experimental data obtained by calorimetry techniques. Also listed is the heat of transformations during cyclic experiments in DSC where a heating/cooling rate of 1◦ C/min was used. Ab initio, J/mola Property η1 →η ∆E tr
0
∆E ηtr2 →η
0
Experimental data (∆E tr ), J/mol 0
CALPHAD (∆E tr ), J/mol 0
η↔η 0
xc-LDA
xc-GGA
Heating (η →η)
-354.1
-320.9
13.8b
-749.1
-609.9
430.9±18c
-396.3±14c
±2604
450.4±12d
-372.2±30d
±3045
395.7±28e
-276.3±29e
Cooling (η→η )
±2603
-263.7±33f Cyclic experimentc Heating
Cooling
1st: 439.4
1st: -349.1
2nd: 388.3
2nd: -327.3
3rd: 351.4
3rd: -308.7
4th: 329.1
4th: -291.1
a This
study study (DSC): as-cast specimen c This study (DSC): specimen annealed at 100 ◦ C d This study (DSC): specimen annealed at 125 ◦ C e This study (DSC): specimen annealed at 150 ◦ C f Gangulee et al.2 (solution calorimetry): specimen annealed at 175 ◦ C b This
η1 and η2 ), and (iii) consistent with the theoretical prediction, experimentally we find that the η 0 ↔η transformation has a very small effect on the bulk modulus. This is also true for other measured elastic constants as mentioned in the previous paragraph.
V. A.
DISCUSSION
Phase stability and phase transformations of Cu6 Sn5
Phase stability: Figure 6 plots the zero-temperature calculated ∆E f as a function of Sn content, in the composition range of Cu2 Sn-CuSn2 . Among the phases considered, B81 -CuSn has the lowest ∆E f . In compositions less than 50 at.% Sn, η and η 0 are more stable compared to phase separated mixture of B82 -Cu2 Sn and B81 -CuSn. In compositions greater than or equal to 50 at.% Sn, the single phase B81 -CuSn lies about 1.8 kJ/mol below the phase separated mixture of η/η 0 and A5-Sn which is in contradiction to the experimental phase diagram where the equiatomic B81 CuSn phase is not observed. A possible reason for this discrepancy could be that entropy differences between the competing phases, η/η 0 and B81 -CuSn, are large enough to reverse their relative stability at finite temperatures. About 0.72 k B /atom entropy difference between η 0 and B81 -CuSn would be required to give rise to the stabilization
18 of former at 300 K. Indeed, the role of vibrational entropy in reconciling a similar apparent discrepancy between theory and experiment for the competing phases of θ-Al2 Cu was demonstrated by Wolverton and Ozolins.66 Calculations of various sources of entropy contributions, (an)harmonic vibrations, electronic and configurational, could possibly reconcile the differences between the zero-temperature ab initio calculations and experimental observations at finite temperatures. Gangulee et al.2 proposed two approaches to calculate the enthalpy of η 0 ↔η transformation, one based on the relative fraction of nearest-neighbor (NN) Sn-Sn bonds relative to the total number of NN bonds affected by the formation of η 0 , and the second approach involves estimation of entropy change associated with the ordering of Cu atoms in tetrahedral sites. However, at the time the crystallographic details of η and η 0 were not known, as a result they estimated the transformation enthalpy based on a specific, but arbitrary, assumption of the number of NN Sn-Sn bonds and the number of atoms in the unit/super cell. Their calculated value of η 0 ↔η transformation lies in the range of 335 to 837 J/mol. Adopting the first approach, but specific to the structural models of η1 and η2 , we find that the heat of transformation lies in the range of 550 to 750 J/mol which is in reasonable agreement with our measured value of 438±18 J/mol.
Figure 5: DSC thermograms after 4 heating/cooling cycles of a specimen annealed at 150 ◦ C for 80 days followed by 100 ◦ C for 60 days. Thermal cycles were carried out between 100 and 220 ◦ C at a rate of 1 ◦ C/min.
Phase transformations: Based on previous experimental studies by various techniques the η 0 ↔ η transformation has been described as being of ferromagnetic,67,68 allotropic,68,69 and order-disorder2,70 type. In addition, in the most recent phase-diagram assessment71 this transformation is indicated by a second-order line. In this section we consider further the nature of this transformation in light of a group/subgroup analysis based on the proposed structural
19
Figure 6: Calculated zero-temperature formation energy (∆E f ) of Cu-Sn intermetallics. Here, the η phase corresponds to the structural model of η1 (see Table I and V). Cu2 Sn (hP 6) and CuSn2 (hP 6) are virtual phases.
models, and the current DSC results described in section IV.C. The Landau theory of phase transitions72 is based on an assumption that the symmetry of the product phase is a subgroup of the parent phase, and that the structures of these phases can be related by a set of order parameters. For a given space group, the maximal group/subgroup relations and the chains of maximal subgroups via an intermediate structure whose space group is either a supergroup of both structures or a subgroup of both structures73 can be obtained from crystallographic databases such as the one found in Ref.74. Table VIII gives the maximal subgroups of η1 (P 21 /c), η2 (C2) and η 0 (C2/c) phases. Also listed in Table VIII is the index, [n], defined by the ratio of the number of symmetry elements in the supergroup and subgroup. In the Landau theory an odd index leads to a first-order transition, while an even index is compatible with either first or second order character. From the information in Table VIII, the space group of the η 0 (C2/c) structure is seen to be a supergroup of both η1 (P 21 /c) and η2 (C2). Since η1 and η2 are models for the high-temperature η phase, this analysis raises the interesting possibility that the η 0 → η could be of order-disorder type. If so, this would represent a novel example of an ordering reaction induced by heating, and would suggest a dominant role of vibrational entropy in governing the transition since the configurational entropy alone would not be expected to stabilize an ordered phase with increasing temperature. We further note, that the even values for the indexes [n] given in Table VIII suggest that this transition could be of second-order type, consistent with the indications given in the phase-diagram assessment. However, the small value of the latent heat and small hysteresis of the η 0 ↔η phase transformation observed in our DSC experiments appears to be incompatible with second-order character and suggest that the transition is weakly first order. Further, we note that given the different stoichiometries of the η (i.e., η1 or η2 ) and η 0 structures, it is not possible to establish a simple set
20 Table VIII: Subgroups of structures relevant to phase transformation of Cu6 Sn5 obtained from a crystallographic database.74 The index, [n], refers to number of possible domain variants. (a) P63 /mmc (#194): B81 -CuSn
(b) P21 /c (#14): η1
Maximal non-isomorphic subgroups
Maximal non-isomorphic subgroups
[3] Cmcm (#63)
[2] P 1 (#2)
[2] P 31c (#163)
[2] P 21 (#4)
[2] P 3m1 (#164)
[2] P c (#7)
[2] P 63 /m (#176)
[2] P 21 /c (#14)
[2] P 63 22 (#182)
[3] P 21 /c (#14)
[2] P 63 mc (#186) [2] P 6m2 (#187) [2] P 62c (#190) [3] P 63 /mcm (#193) [3] P 63 /mmc (#194) [4] P 63 /mmc (#194)
(c) C 2 (#5): η2
(d) C 2/c (#15): η 0
Maximal non-isomorphic subgroups
Maximal non-isomorphic subgroups
[2] P 1 (#1)
[2] P 1 (#2)
[2] P 2 (#3)
[2] C2 (#5)
[2] P 21 (#4)
[2] Cc (#9)
[2] C2 (#5)
[2] P 2/c (#13)
[3] C2 (#5)
[2] P 21 /c (#14) [3] C2/c (#15)
of order parameters that would take one phase into the other. Additional research utilizing diffraction technique(s) is required to firmly establish the nature of η↔η 0 phase transformation. Specifically, (i) structural models of η and η 0 could be developed at the same stoichiometry which is currently lacking, (ii) information related to the sublattice occupations for the off-stoichiometric phases is required, and (iii) a method which could establish whether η1 or η2 is the more accurate representation of the high-temperature η phase. Finally, we comment on the observations of the DSC experiments which suggest two peaks on cooling. Based on the calculated energy results plotted in Fig. 2 it is possible that the these peaks could be representative of a transformation sequence η2 → η1 → η 0 upon cooling. In such a scenario, the lack of two peaks upon heating would imply that the η1 is a transient phases which forms upon cooling for kinetic reasons. Once again, to determine the exact nature of the structural transitions, our DSC results strongly suggest that further in-situ studies, preferably by X-Ray diffraction, are warranted to determine the exact details of the transformation sequence upon heating and cooling.
21 B.
Elastic properties of Cu6 Sn5
As mentioned before, there have been several attempts to measure the elastic properties of Cu6 Sn5 , but the data show a large scatter. Like the present study, Subrahmanyan13 also used bulk specimens and measured the shear and Youngs moduli of Cu6 Sn5 by frequency resonance technique. Fields et al.15 prepared bulk specimens of Cu6 Sn5 by powder metallurgy technique. They measured the Youngs modulus and Poissons ratio by compression testing, and then derived the shear modulus with the assumption of isotropic behavior of the specimens. Using Eq. 8, the calculated bulk modulus is 14.8×1010 N/m2 based on Subrahmanyan’s13 data, and 7.46×1010 N/m2 based on Fields15 data. These are about 75% higher and 13% lower, respectively, compared to our measured value of 8.46 × 1010 N/m2 by pulse-echo technique. A very high value of bulk modulus using Subrahmanyan’s data13 arises due to relatively low values of shear and Young’s moduli via Eq. (8). A closer examination of his data shows that the measured shear and Youngs moduli of pure Cu and Sn are systematically lower than the compiled values.50 Furthermore, Subrahmanyan13 annealed the alloys at 473 K for only 10 hrs, and did not show any evidence of single phase microstructure. Therefore, we believe that there is a systematic error associated with his measurement technique. An unexpected result is the decrease in bulk modulus (both theoretical and experimental) of η1 , η2 and η 0 relative to a concentration-weighted average of the pure-element values. A negative heat of formation of these phases implies favorable chemical bonding between Cu and Sn atoms. The EOS parameters listed in Tables II and IV demonstrate the decreases of both mean atomic volume and bulk modulus relative to the composition-average pure-element values. While the former is expected, the latter is expected to be otherwise. This observation is also true for a simple structure like B81 -CuSn. Furthermore, our observation is true irrespective of whether the comparison is made based on LDAor GGA-level calculated values, or the experimental values. We do not know the origin of this unexpected result; however, it may be related to unusual behavior of Sn with regard to bulk modulus and its temperature dependence. Unlike the normal behavior, the bulk modulus of Sn decreases with decreasing temperature.50 Compilation of Young’s modulus data13–15,17–20 shows a spread of 8.43 to 11.9 × 1010 N/m2 . It is important to note that due to monoclinic symmetry of η and η 0 phases, Young’s modulus will be anisotropic. Therefore, the measured values will depend on the type of specimen and the measurement technique. A combination of these two factors determines the number of orientations being sampled during measurement. In fact, experimental Young’s modulus measurement using bulk specimens are systematically lower, being in the range of 8.43 to 10.24 × 1010 N/m2 ,13–15 compared to the values obtained by nanoindentation techniques using Cu6 Sn5 -layer in diffusion couples17–19 that lie in the range of 10.8 to 11.9 × 1010 N/m2 . It is true that the nanoindentation technique requires a small specimen to determine Young’s modulus, but the technique suffers from at least two drawbacks: (i) it requires a prior knowledge of Poissons ratio (measured independently), and (ii) the results may be biased if there is a preferred orientation of the growing intermetallic layer(s) in diffusion couples. In the case of Cu6 Sn5 -layer in diffusion couples, Prakash and Sritharan75,76 observed a h102i and h101i texture. Therefore, we believe that the Young’s modulus obtained by nanoindentation technique and using a layer of Cu6 Sn5 in diffusion couples most likely corresponds to a particular or at most few orientations, and do not reflect a property averaged over many orientations. Our measured Young’s modulus of 9.69 × 1010 N/m2 for η and 9.44 × 1010 N/m2 for η 0 are somewhat close to the average of all previously reported values, which is 10.17 × 1010 N/m2 . Due to monoclinic symmetry of η and η 0 phases and fairly large unit cells, the calculation of Young’s modulus by ab initio techniques is computationally very
22 demanding. However, we believe that since our experimental bulk modulus agrees very well with the US-PP-GGA values, other measured moduli (shear, Young’s and Poissons ratio) are also representative of actual isotropic properties of Cu6 Sn5 . This is due to the fact that in pulse-echo technique only two fundamental quantities, the velocity of shear and longitudinal waves, are measured directly. As seen in Eq. 6 to 9, all isotropic elastic constants can be expressed in terms of these two quantities.
VI.
CONCLUSIONS
The phase stability and elastic properties of Cu and Sn with six structures, and of Cu6 Sn5 with structures reported in the literature, are calculated from first-principles using electronic density-functional theory. All ab initio calculations are performed using US-PP, and both LDA and GGA levels as implemented in VASP. Two critical experiments using Cu6 Sn5 , measurement of heat of transformation by differential scanning calorimetry and bulk modulus by pulse-echo technique, are performed to validate some of our ab initio predictions. The following conclusions are drawn: For Cu and Sn with simple structures, such as bcc, fcc and hcp, the calculated zero-temperature lattice stabilities agree fairly well with the data available in the SGTE database. We provide the calculated lattice stabilities of Cu and Sn in A4, A5 and A6 structures and compare them with the previously reported ab initio results. For Cu, VASP-GGA results for lattice parameter and bulk modulus agreement within ±1.3% when compared with the corresponding experimental data at 0 K. For Sn, VASP-LDA results for lattice parameter and bulk modulus are in very good agreement with the experimental data compared to VASP-GGA, but VASP-LDA fails to predict the correct ground state structure. For η 0 -Cu6 Sn5 , the calculated formation energy lies between -3.2 to -4.0 kJ/mol which is more positive by about 3 to 4 kJ/mol compared to previously reported experimental data obtained by solution calorimetry. Our measured η 0 →η transformation enthalpy is 438±18 J/mol, which is larger by a factor of 1.66 compared with the previously reported value. Comparing the ab initio structural energetics of η1 , η2 , and η 0 and their elastic properties with our experimental η 0 ↔η transformation enthalpy and the bulk modulus data, we find that US-PP-GGA gives better overall agreement. Consistent with our experimental measurements, ab initio results also show small differences in the bulk moduli of η1 , η2 , and η 0 structures. Due to small hysteresis and latent heat, η 0 ↔η phase transformation is considered to be weakly first-order. This study demonstrates that phase transformations involving a small change in enthalpy, less than 500 J/mol, can be calculated by modern DFT techniques and also be readily verified by a simple experimental technique such as DSC to validate crystallographic structural models.
Acknowledgments
This work was supported by the National Science Foundation under the Grant# DMR-9813919. Supercomputing resources were provided by the National Partnership for Advanced Computational Infrastructure (NPACI) at the University of Michigan and at the University of Illinois at Urbana-Champaign. The authors would like to thank the following individuals: Michele Manuel, Northwestern University, for her help with DSC experiments; Prof. Georg
23 Kresse, Universit¨ at Wien, for helpful comments relevant to the VASP calculations for Sn; and Dr. Avadh Saxena, Los Alamos National Laboratory, for helpful discussion.
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